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The epigraph is $\{(u,t):E(u)\le t\}$. It is a convex set exactly when $E$ is a convex function, and is a sequentially closed set in the product topology exactly when $E$ is sequentially lower semicontinuous.

The epigraph isAssume $E$ is $\tau$-sequentially lower semicontinuous. If $(u_n,t_n)$ lies in the epigraph and converges to $(u,t)$ in the product topology, thenThe limit remains in the epigraph, so it is a sequentially closed set.

Conversely, suppose the epigraph is a sequentially closed set and sequential lower semicontinuity fails along $u_n\stackrel{\tau }{\to }u$. Choose a finite real number $a$ with $\mathrm{liminf}E(u_n)<a<E(u)$. There is a subsequence along which $E(u_{n_j})\le a$. Thus $(u_{n_j},a)$ lies in the epigraph and converges to $(u,a)$, which does not lie there. This contradiction provesThis is a sequential statement; identifying it with ordinary closedness requires an appropriate assumption on the topology.

A sequentially closed set is one that contains every limit of its convergent sequences. This is weaker than being a closed set in a general topological space, but the two conditions agree in metric spaces.

A sublevel set consists of points whose objective value is at most a chosen threshold. Coercivity makes finite sublevel sets bounded, while sequential lower semicontinuity makes them sequentially closed sets in the selected topology.